288 research outputs found
Deformed Statistics Kullback-Leibler Divergence Minimization within a Scaled Bregman Framework
The generalized Kullback-Leibler divergence (K-Ld) in Tsallis statistics
[constrained by the additive duality of generalized statistics (dual
generalized K-Ld)] is here reconciled with the theory of Bregman divergences
for expectations defined by normal averages, within a measure-theoretic
framework. Specifically, it is demonstrated that the dual generalized K-Ld is a
scaled Bregman divergence. The Pythagorean theorem is derived from the minimum
discrimination information-principle using the dual generalized K-Ld as the
measure of uncertainty, with constraints defined by normal averages. The
minimization of the dual generalized K-Ld, with normal averages constraints, is
shown to exhibit distinctly unique features.Comment: 16 pages. Iterative corrections and expansion
Nonextensive statistics: Theoretical, experimental and computational evidences and connections
The domain of validity of standard thermodynamics and Boltzmann-Gibbs
statistical mechanics is discussed and then formally enlarged in order to
hopefully cover a variety of anomalous systems. The generalization concerns
{\it nonextensive} systems, where nonextensivity is understood in the
thermodynamical sense. This generalization was first proposed in 1988 inspired
by the probabilistic description of multifractal geometries, and has been
intensively studied during this decade. In the present effort, after
introducing some historical background, we briefly describe the formalism, and
then exhibit the present status in what concerns theoretical, experimental and
computational evidences and connections, as well as some perspectives for the
future. In addition to these, here and there we point out various (possibly)
relevant questions, whose answer would certainly clarify our current
understanding of the foundations of statistical mechanics and its
thermodynamical implicationsComment: 15 figure
Spectral Clustering with Jensen-type kernels and their multi-point extensions
Motivated by multi-distribution divergences, which originate in information
theory, we propose a notion of `multi-point' kernels, and study their
applications. We study a class of kernels based on Jensen type divergences and
show that these can be extended to measure similarity among multiple points. We
study tensor flattening methods and develop a multi-point (kernel) spectral
clustering (MSC) method. We further emphasize on a special case of the proposed
kernels, which is a multi-point extension of the linear (dot-product) kernel
and show the existence of cubic time tensor flattening algorithm in this case.
Finally, we illustrate the usefulness of our contributions using standard data
sets and image segmentation tasks.Comment: To appear in IEEE Computer Society Conference on Computer Vision and
Pattern Recognitio
Generalized Statistics Variational Perturbation Approximation using q-Deformed Calculus
A principled framework to generalize variational perturbation approximations
(VPA's) formulated within the ambit of the nonadditive statistics of Tsallis
statistics, is introduced. This is accomplished by operating on the terms
constituting the perturbation expansion of the generalized free energy (GFE)
with a variational procedure formulated using \emph{q-deformed calculus}. A
candidate \textit{q-deformed} generalized VPA (GVPA) is derived with the aid of
the Hellmann-Feynman theorem. The generalized Bogoliubov inequality for the
approximate GFE are derived for the case of canonical probability densities
that maximize the Tsallis entropy. Numerical examples demonstrating the
application of the \textit{q-deformed} GVPA are presented. The qualitative
distinctions between the \textit{q-deformed} GVPA model \textit{vis-\'{a}-vis}
prior GVPA models are highlighted.Comment: 26 pages, 4 figure
Learning from Distributions via Support Measure Machines
This paper presents a kernel-based discriminative learning framework on
probability measures. Rather than relying on large collections of vectorial
training examples, our framework learns using a collection of probability
distributions that have been constructed to meaningfully represent training
data. By representing these probability distributions as mean embeddings in the
reproducing kernel Hilbert space (RKHS), we are able to apply many standard
kernel-based learning techniques in straightforward fashion. To accomplish
this, we construct a generalization of the support vector machine (SVM) called
a support measure machine (SMM). Our analyses of SMMs provides several insights
into their relationship to traditional SVMs. Based on such insights, we propose
a flexible SVM (Flex-SVM) that places different kernel functions on each
training example. Experimental results on both synthetic and real-world data
demonstrate the effectiveness of our proposed framework.Comment: Advances in Neural Information Processing Systems 2
- …